Tugas Week 4

# Komputasi Statistika, Kelas D Prodi S1 Statistika FSAD ITS - Semester Genap
# 2025/2026 Selasa, 17 Maret 2026
# 
# Nama  : Andin Olga Pramesti
# NRP   : 5003251065
# Kelas : C

Soal 1 - Winsorized Mean

# Data
x <- c(12, 45, 52, 58, 61, 63, 67, 70, 72, 75, 78, 82, 88, 95, 310)

# [a] Buatlah fungsi winsorized_mean(x, alpha)
winsorized_mean <- function(x, alpha){
  X <- sort(x)
  n <- length(X)
  k <- floor(n*alpha)
  
  y <- NULL

  for(i in 1:n){
    if(i<=k){
      y[i] <- X[k+1]
    } else if(k<i & i<=(n-k)){
      y[i] <- X[i]
    } else if(i>(n-k)){
      y[i] <- X[n-k] 
    }
  }
  
  jumlah <- 0
  for(i in 1:n){
    jumlah <- jumlah + y[i]
  }
  
  return(jumlah/n)
}

# [b] Hitung ordinary mean (alpha=0)
winsorized_mean(x, 0)

[1] 81.86667

# Winsorized mean 20% (alpha=0.2)
winsorized_mean(x, 0.2)

[1] 69.73333

ordinary_mean <- winsorized_mean(x, 0)
winsor_mean <- winsorized_mean(x, 0.2)

boxplot(x,
        main = "Output Produksi Mesin",
        ylab = "Output (Per Hari)",
        col = "yellow")

abline(h = ordinary_mean,
       col="orange", 
       lwd=2)

abline(h = winsor_mean,
       col="red", 
       lwd=2)

legend("topright",
       legend=c("Ordinary Mean", "Winsorized Mean"),
       col=c("orange", "red"),
       lwd=2,
       cex = 0.8)

ANALISIS DAN INTERPRETASI

Dari boxplot tersebut terlihat bahwa terdapat satu titik ekstrem di bagian atas serta satu titik di bagian bawah, sehingga data tersebut dapat diindikasikan mengandung outlier.

Nilai rata-rata biasa (ordinary mean) yang diperoleh adalah sebesar 81,87, sedangkan winsorized mean dengan alpha = 0,2 adalah sebesar 69,73. Perbedaan ini menunjukkan bahwa rata-rata biasa dipengaruhi oleh nilai ekstrem, khususnya outlier sebesar 310, sehingga nilainya menjadi lebih tinggi dari pusat data sebenarnya.

Setelah dilakukan winsorizing, nilai ekstrem tersebut disesuaikan sehingga pengaruhnya berkurang. Akibatnya, winsorized mean menjadi lebih rendah dan lebih mendekati nilai mayoritas data.

Soal 2 - Weighted Multivariate Descriptive Statistics

# Baca data CSV
df <- read.csv("data_quiz1.csv")
X <- as.matrix(df[, c("x1", "x2", "x3")])
w <- df$w

# [a] Buatlah fungsi 'weighted_corr(X, w)
weighted_corr <- function(X, w) {
  X <- as.matrix(X)
  
  W <- diag(w)
  n <- nrow(X)
  m1 <- rep(1, times=n)
  
  nw <- 0
  for(i in 1:n){
    nw <- nw + w[i]
  }
  
  x_bar_w <- (t(X) %*% W %*% m1)/nw
  
  D <- X - m1 %*% t(x_bar_w)
  
  S_w <- (t(D) %*% W %*% D)/nw
  
  s_w <- sqrt(diag(S_w))
  
  V <- diag(s_w)
  
  R_w <- solve(V) %*% S_w %*% solve(V)
  
  return(list(
    W = W,
    x_bar_w = x_bar_w,
    S_w = S_w,
    s_w = s_w,
    R_w = R_w
  ))
}

# [b] Aplikasikan fungsi pada data
# Panggil fungsi
weighted_corr(X, w)

$W
       [,1]  [,2]  [,3]  [,4]  [,5]
 [1,] 14.34  0.00  0.00  0.00  0.00
 [2,]  0.00 14.19  0.00  0.00  0.00
 [3,]  0.00  0.00 12.49  0.00  0.00
 [4,]  0.00  0.00  0.00 11.45  0.00
 [5,]  0.00  0.00  0.00  0.00 17.45
 [6,]  0.00  0.00  0.00  0.00  0.00
 [7,]  0.00  0.00  0.00  0.00  0.00
 [8,]  0.00  0.00  0.00  0.00  0.00
 [9,]  0.00  0.00  0.00  0.00  0.00
[10,]  0.00  0.00  0.00  0.00  0.00
[11,]  0.00  0.00  0.00  0.00  0.00
[12,]  0.00  0.00  0.00  0.00  0.00
[13,]  0.00  0.00  0.00  0.00  0.00
[14,]  0.00  0.00  0.00  0.00  0.00
[15,]  0.00  0.00  0.00  0.00  0.00
[16,]  0.00  0.00  0.00  0.00  0.00
[17,]  0.00  0.00  0.00  0.00  0.00
[18,]  0.00  0.00  0.00  0.00  0.00
[19,]  0.00  0.00  0.00  0.00  0.00
[20,]  0.00  0.00  0.00  0.00  0.00
[21,]  0.00  0.00  0.00  0.00  0.00
[22,]  0.00  0.00  0.00  0.00  0.00
[23,]  0.00  0.00  0.00  0.00  0.00
[24,]  0.00  0.00  0.00  0.00  0.00
[25,]  0.00  0.00  0.00  0.00  0.00
[26,]  0.00  0.00  0.00  0.00  0.00
       [,6]  [,7]  [,8]  [,9] [,10]
 [1,]  0.00  0.00  0.00  0.00  0.00
 [2,]  0.00  0.00  0.00  0.00  0.00
 [3,]  0.00  0.00  0.00  0.00  0.00
 [4,]  0.00  0.00  0.00  0.00  0.00
 [5,]  0.00  0.00  0.00  0.00  0.00
 [6,] 15.24  0.00  0.00  0.00  0.00
 [7,]  0.00 34.73  0.00  0.00  0.00
 [8,]  0.00  0.00 17.97  0.00  0.00
 [9,]  0.00  0.00  0.00 33.13  0.00
[10,]  0.00  0.00  0.00  0.00 35.93
[11,]  0.00  0.00  0.00  0.00  0.00
[12,]  0.00  0.00  0.00  0.00  0.00
[13,]  0.00  0.00  0.00  0.00  0.00
[14,]  0.00  0.00  0.00  0.00  0.00
[15,]  0.00  0.00  0.00  0.00  0.00
[16,]  0.00  0.00  0.00  0.00  0.00
[17,]  0.00  0.00  0.00  0.00  0.00
[18,]  0.00  0.00  0.00  0.00  0.00
[19,]  0.00  0.00  0.00  0.00  0.00
[20,]  0.00  0.00  0.00  0.00  0.00
[21,]  0.00  0.00  0.00  0.00  0.00
[22,]  0.00  0.00  0.00  0.00  0.00
[23,]  0.00  0.00  0.00  0.00  0.00
[24,]  0.00  0.00  0.00  0.00  0.00
[25,]  0.00  0.00  0.00  0.00  0.00
[26,]  0.00  0.00  0.00  0.00  0.00
      [,11] [,12] [,13] [,14] [,15]
 [1,]  0.00  0.00  0.00  0.00  0.00
 [2,]  0.00  0.00  0.00  0.00  0.00
 [3,]  0.00  0.00  0.00  0.00  0.00
 [4,]  0.00  0.00  0.00  0.00  0.00
 [5,]  0.00  0.00  0.00  0.00  0.00
 [6,]  0.00  0.00  0.00  0.00  0.00
 [7,]  0.00  0.00  0.00  0.00  0.00
 [8,]  0.00  0.00  0.00  0.00  0.00
 [9,]  0.00  0.00  0.00  0.00  0.00
[10,]  0.00  0.00  0.00  0.00  0.00
[11,] 15.55  0.00  0.00  0.00  0.00
[12,]  0.00 16.54  0.00  0.00  0.00
[13,]  0.00  0.00 17.25  0.00  0.00
[14,]  0.00  0.00  0.00 14.93  0.00
[15,]  0.00  0.00  0.00  0.00  7.24
[16,]  0.00  0.00  0.00  0.00  0.00
[17,]  0.00  0.00  0.00  0.00  0.00
[18,]  0.00  0.00  0.00  0.00  0.00
[19,]  0.00  0.00  0.00  0.00  0.00
[20,]  0.00  0.00  0.00  0.00  0.00
[21,]  0.00  0.00  0.00  0.00  0.00
[22,]  0.00  0.00  0.00  0.00  0.00
[23,]  0.00  0.00  0.00  0.00  0.00
[24,]  0.00  0.00  0.00  0.00  0.00
[25,]  0.00  0.00  0.00  0.00  0.00
[26,]  0.00  0.00  0.00  0.00  0.00
      [,16] [,17] [,18] [,19] [,20]
 [1,]  0.00   0.0  0.00  0.00  0.00
 [2,]  0.00   0.0  0.00  0.00  0.00
 [3,]  0.00   0.0  0.00  0.00  0.00
 [4,]  0.00   0.0  0.00  0.00  0.00
 [5,]  0.00   0.0  0.00  0.00  0.00
 [6,]  0.00   0.0  0.00  0.00  0.00
 [7,]  0.00   0.0  0.00  0.00  0.00
 [8,]  0.00   0.0  0.00  0.00  0.00
 [9,]  0.00   0.0  0.00  0.00  0.00
[10,]  0.00   0.0  0.00  0.00  0.00
[11,]  0.00   0.0  0.00  0.00  0.00
[12,]  0.00   0.0  0.00  0.00  0.00
[13,]  0.00   0.0  0.00  0.00  0.00
[14,]  0.00   0.0  0.00  0.00  0.00
[15,]  0.00   0.0  0.00  0.00  0.00
[16,]  9.85   0.0  0.00  0.00  0.00
[17,]  0.00  11.1  0.00  0.00  0.00
[18,]  0.00   0.0 12.89  0.00  0.00
[19,]  0.00   0.0  0.00 11.14  0.00
[20,]  0.00   0.0  0.00  0.00  7.06
[21,]  0.00   0.0  0.00  0.00  0.00
[22,]  0.00   0.0  0.00  0.00  0.00
[23,]  0.00   0.0  0.00  0.00  0.00
[24,]  0.00   0.0  0.00  0.00  0.00
[25,]  0.00   0.0  0.00  0.00  0.00
[26,]  0.00   0.0  0.00  0.00  0.00
      [,21] [,22] [,23] [,24] [,25]
 [1,]  0.00  0.00  0.00  0.00  0.00
 [2,]  0.00  0.00  0.00  0.00  0.00
 [3,]  0.00  0.00  0.00  0.00  0.00
 [4,]  0.00  0.00  0.00  0.00  0.00
 [5,]  0.00  0.00  0.00  0.00  0.00
 [6,]  0.00  0.00  0.00  0.00  0.00
 [7,]  0.00  0.00  0.00  0.00  0.00
 [8,]  0.00  0.00  0.00  0.00  0.00
 [9,]  0.00  0.00  0.00  0.00  0.00
[10,]  0.00  0.00  0.00  0.00  0.00
[11,]  0.00  0.00  0.00  0.00  0.00
[12,]  0.00  0.00  0.00  0.00  0.00
[13,]  0.00  0.00  0.00  0.00  0.00
[14,]  0.00  0.00  0.00  0.00  0.00
[15,]  0.00  0.00  0.00  0.00  0.00
[16,]  0.00  0.00  0.00  0.00  0.00
[17,]  0.00  0.00  0.00  0.00  0.00
[18,]  0.00  0.00  0.00  0.00  0.00
[19,]  0.00  0.00  0.00  0.00  0.00
[20,]  0.00  0.00  0.00  0.00  0.00
[21,] 13.96  0.00  0.00  0.00  0.00
[22,]  0.00 23.13  0.00  0.00  0.00
[23,]  0.00  0.00 19.74  0.00  0.00
[24,]  0.00  0.00  0.00 17.53  0.00
[25,]  0.00  0.00  0.00  0.00 12.56
[26,]  0.00  0.00  0.00  0.00  0.00
      [,26] [,27] [,28] [,29] [,30]
 [1,]  0.00  0.00  0.00  0.00  0.00
 [2,]  0.00  0.00  0.00  0.00  0.00
 [3,]  0.00  0.00  0.00  0.00  0.00
 [4,]  0.00  0.00  0.00  0.00  0.00
 [5,]  0.00  0.00  0.00  0.00  0.00
 [6,]  0.00  0.00  0.00  0.00  0.00
 [7,]  0.00  0.00  0.00  0.00  0.00
 [8,]  0.00  0.00  0.00  0.00  0.00
 [9,]  0.00  0.00  0.00  0.00  0.00
[10,]  0.00  0.00  0.00  0.00  0.00
[11,]  0.00  0.00  0.00  0.00  0.00
[12,]  0.00  0.00  0.00  0.00  0.00
[13,]  0.00  0.00  0.00  0.00  0.00
[14,]  0.00  0.00  0.00  0.00  0.00
[15,]  0.00  0.00  0.00  0.00  0.00
[16,]  0.00  0.00  0.00  0.00  0.00
[17,]  0.00  0.00  0.00  0.00  0.00
[18,]  0.00  0.00  0.00  0.00  0.00
[19,]  0.00  0.00  0.00  0.00  0.00
[20,]  0.00  0.00  0.00  0.00  0.00
[21,]  0.00  0.00  0.00  0.00  0.00
[22,]  0.00  0.00  0.00  0.00  0.00
[23,]  0.00  0.00  0.00  0.00  0.00
[24,]  0.00  0.00  0.00  0.00  0.00
[25,]  0.00  0.00  0.00  0.00  0.00
[26,] 13.01  0.00  0.00  0.00  0.00
      [,31] [,32] [,33] [,34] [,35]
 [1,]  0.00  0.00  0.00  0.00   0.0
 [2,]  0.00  0.00  0.00  0.00   0.0
 [3,]  0.00  0.00  0.00  0.00   0.0
 [4,]  0.00  0.00  0.00  0.00   0.0
 [5,]  0.00  0.00  0.00  0.00   0.0
 [6,]  0.00  0.00  0.00  0.00   0.0
 [7,]  0.00  0.00  0.00  0.00   0.0
 [8,]  0.00  0.00  0.00  0.00   0.0
 [9,]  0.00  0.00  0.00  0.00   0.0
[10,]  0.00  0.00  0.00  0.00   0.0
[11,]  0.00  0.00  0.00  0.00   0.0
[12,]  0.00  0.00  0.00  0.00   0.0
[13,]  0.00  0.00  0.00  0.00   0.0
[14,]  0.00  0.00  0.00  0.00   0.0
[15,]  0.00  0.00  0.00  0.00   0.0
[16,]  0.00  0.00  0.00  0.00   0.0
[17,]  0.00  0.00  0.00  0.00   0.0
[18,]  0.00  0.00  0.00  0.00   0.0
[19,]  0.00  0.00  0.00  0.00   0.0
[20,]  0.00  0.00  0.00  0.00   0.0
[21,]  0.00  0.00  0.00  0.00   0.0
[22,]  0.00  0.00  0.00  0.00   0.0
[23,]  0.00  0.00  0.00  0.00   0.0
[24,]  0.00  0.00  0.00  0.00   0.0
[25,]  0.00  0.00  0.00  0.00   0.0
[26,]  0.00  0.00  0.00  0.00   0.0
      [,36] [,37] [,38]
 [1,]  0.00  0.00  0.00
 [2,]  0.00  0.00  0.00
 [3,]  0.00  0.00  0.00
 [4,]  0.00  0.00  0.00
 [5,]  0.00  0.00  0.00
 [6,]  0.00  0.00  0.00
 [7,]  0.00  0.00  0.00
 [8,]  0.00  0.00  0.00
 [9,]  0.00  0.00  0.00
[10,]  0.00  0.00  0.00
[11,]  0.00  0.00  0.00
[12,]  0.00  0.00  0.00
[13,]  0.00  0.00  0.00
[14,]  0.00  0.00  0.00
[15,]  0.00  0.00  0.00
[16,]  0.00  0.00  0.00
[17,]  0.00  0.00  0.00
[18,]  0.00  0.00  0.00
[19,]  0.00  0.00  0.00
[20,]  0.00  0.00  0.00
[21,]  0.00  0.00  0.00
[22,]  0.00  0.00  0.00
[23,]  0.00  0.00  0.00
[24,]  0.00  0.00  0.00
[25,]  0.00  0.00  0.00
[26,]  0.00  0.00  0.00
 [ reached 'max' / getOption("max.print") -- omitted 12 rows ]

$x_bar_w
       [,1]
x1 73.88530
x2 65.39059
x3 17.00938

$S_w
          x1        x2        x3
x1  38.16362 -37.75105 -27.15386
x2 -37.75105  41.10767  29.16587
x3 -27.15386  29.16587  21.14757

$s_w
      x1       x2       x3 
6.177671 6.411527 4.598649 

$R_w
           [,1]       [,2]       [,3]
[1,]  1.0000000 -0.9531095 -0.9558207
[2,] -0.9531095  1.0000000  0.9891979
[3,] -0.9558207  0.9891979  1.0000000

# Tampilkan vektor mean tertimbang
weighted_corr(X, w)$x_bar_w

       [,1]
x1 73.88530
x2 65.39059
x3 17.00938

# Tampilkan matriks varians-kovarians tertimbang
weighted_corr(X, w)$S_w

          x1        x2        x3
x1  38.16362 -37.75105 -27.15386
x2 -37.75105  41.10767  29.16587
x3 -27.15386  29.16587  21.14757

# Tampilkan vektor standar deviasi tertimbang
weighted_corr(X, w)$s_w

      x1       x2       x3 
6.177671 6.411527 4.598649 

# Tampilkan matriks korelasi tertimbang
weighted_corr(X, w)$R_w

           [,1]       [,2]       [,3]
[1,]  1.0000000 -0.9531095 -0.9558207
[2,] -0.9531095  1.0000000  0.9891979
[3,] -0.9558207  0.9891979  1.0000000

# VISUALISASI MATRIKS X
install.packages("GGally")

WARNING: Rtools is required to build R packages but is not currently installed. Please download and install the appropriate version of Rtools before proceeding:

https://cran.rstudio.com/bin/windows/Rtools/
trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.5/GGally_2.4.0.zip'
Content type 'application/zip' length 2058993 bytes (2.0 MB)
downloaded 2.0 MB

package ‘GGally’ successfully unpacked and MD5 sums checked

The downloaded binary packages are in
    C:\Users\LENOVO\AppData\Local\Temp\RtmpaqsHbM\downloaded_packages

library(GGally)
library(ggplot2)

ggpairs(df[,2:4])

ANALISIS DAN INTERPRETASI

Secara umum pair plot terdiri dari 3 komponen utama, yaitu scatter plot, korelasi, dan distribusi antar variabel yang disajikan melalui density plot.

Berdasarkan scatter plot dan nilai korelasi yang ditampilkan pada pair plot, dapat disimpulkan bahwa variabel x1 dan x2 memiliki korelasi negatif yang sangat kuat (sekitar -0,969), yang menunjukkan adanya hubungan terbalik antara keduanya. Hal serupa juga terlihat pada hubungan antara x1 dan x3, yang memiliki korelasi negatif sangat kuat (sekitar -0,975), sehingga menunjukkan pola hubungan terbalik. Sementara itu, variabel x2 dan x3 memiliki korelasi positif yang sangat kuat (sekitar 0,991), yang mengindikasikan bahwa kedua variabel tersebut memiliki hubungan yang berbanding lurus.

Selain itu, pair plot juga menampilkan distribusi masing-masing variabel melalui density plot. Berdasarkan visualisasi tersebut, terlihat bahwa distribusi variabel x1, x2, dan x3 tidak sepenuhnya simetris, yang mengindikasikan adanya sedikit kemencengan (skewness) pada data